As defined by Rattay, in the article, “Analysis of models for extracellular fiber stimulation,” IEEE Transactions on Biomedical Engineering, Vol. 36, no. 2, p. 676, 1989, which is incorporated herein by reference, the activation function (AF) is the second spatial derivative of the electric potential along an axon. In the region where the activation function is positive, the axon depolarizes, and in the region where the activation function is negative, the axon hyperpolarizes. If the activation function is sufficiently positive, then the depolarization will cause the axon to generate an action potential; similarly, if the activation function is sufficiently negative, then local blocking of action potentials transmission occurs. The activation function depends on the current applied, as well as the geometry of the electrodes and of the axon.
For a given electrode geometry, the equation governing the electrical potential is:∇(σ∇U)=4πj, 
where U is the potential, σ is the conductance tensor specifying the conductance of the various materials (electrode housing, axon, intracellular fluid, etc.), and j is a scalar function representing the current source density specifying the locations of current injection. The activation function is found by solving this partial differential equation for U. If the axon is defined to lie in the z direction, then the activation function is:
  AF  =                              ∂          2                ⁢        U                    ∂                  z          2                      .  
In a simple, illustrative example of a point electrode located a distance d from the axis of an axon in a uniformly-conducting medium with conductance σ, the two equations above are solvable analytically, to yield:
      AF    =                            I          e1                          4          ⁢          πσ                    ·                                    2            ⁢                          z              2                                -                      d            2                                                (                                          z                2                            +                              d                2                                      )                    2.5                      ,
where Iel is the electrode current. It is seen that when σ and d are held constant, and for a constant positive Iel (to correspond to anodal current), the minimum value of the activation function is negative, and is attained at z=0, i.e., at the point on the nerve closest to the source of the anodal current. Thus, the most negative point on the activation function corresponds to the place on a nerve where hyperpolarization is maximized, namely at the point on the nerve closest to the anode.
Additionally, this equation predicts positive “lobes” for the activation function on either side of z=0, these positive lobes peaking in their values at a distance which is dependent on each of the other parameters in the equation. The positive values of the activation function correspond to areas of depolarization, a phenomenon typically associated with cathodic current, not anodal current. However, it has been shown that excess anodal current does indeed cause the generation of action potentials adjacent to the point on a nerve corresponding to z=0, and this phenomenon is therefore called the “virtual cathode effect.” (An analogous, but reverse phenomenon, the “virtual anode effect” exists responsive to excess cathodic stimulation.)
U.S. Pat. No. 6,230,061 to Hartung, which is incorporated herein by reference, describes an electrode arrangement for stimulating the heart by means of: (a) an implantable cardiac pacemaker, (b) a first electrode, coupled to a first output of the pacemaker via an intracardiac electrode line, and (c) a second electrode, for transmitting electrical stimulation pulses to the heart tissue, coupled to a second output of the pacemaker via the electrode line. The voltage pulses at the two electrodes have differing polarities relative to a third electrode. The first and second electrodes are arranged on the electrode line in such a way that the electrical dipole field which forms is distorted towards the stimulation point in such a way that a raised gradient above the stimulus threshold is formed there.
A number of patents and articles describe methods and devices for stimulating nerves to achieve a desired effect. Often these techniques include a design for an electrode or electrode cuff.
U.S. Pat. Nos. 4,608,985 to Crish et al. and 4,649,936 to Ungar et al., which are incorporated herein by reference, describe electrode cuffs for selectively blocking orthodromic action potentials passing along a nerve trunk, in a manner intended to avoid causing nerve damage.
PCT Patent Publication WO 01/10375 to Felsen et al., which is incorporated herein by reference, describes apparatus for modifying the electrical behavior of nervous tissue. Electrical energy is applied with an electrode to a nerve in order to selectively inhibit propagation of an action potential.
U.S. Pat. No. 5,755,750 to Petruska et al., which is incorporated herein by reference, describes techniques for selectively blocking different size fibers of a nerve by applying direct electric current between an anode and a cathode that is larger than the anode.
The following articles, which are incorporated herein by reference, may be of interest:
Ungar I J et al., “Generation of unidirectionally propagating action potentials using a monopolar electrode cuff,” Annals of Biomedical Engineering, 14:437-450 (1986)
Sweeney J D et al., “An asymmetric two electrode cuff for generation of unidirectionally propagated action potentials,” IEEE Transactions on Biomedical Engineering, vol. BME-33(6) (1986)
Sweeney J D et al., “A nerve cuff technique for selective excitation of peripheral nerve trunk regions,” IEEE Transactions on Biomedical Engineering, 37(7) (1990)
Naples G G et al., “A spiral nerve cuff electrode for peripheral nerve stimulation,” by IEEE Transactions on Biomedical Engineering, 35(11) (1988)
van den Honert C et al., “Generation of unidirectionally propagated action potentials in a peripheral nerve by brief stimuli,” Science, 206:1311-1312 (1979)
van den Honert C et al., “A technique for collision block of peripheral nerve: Single stimulus analysis,” MP-11, IEEE Trans. Biomed. Eng. 28:373-378 (1981)
van den Honert C et al., “A technique for collision block of peripheral nerve: Frequency dependence,” MP-12, IEEE Trans. Biomed. Eng. 28:379-382 (1981)
Rijkhoff N J et al., “Acute animal studies on the use of anodal block to reduce urethral resistance in sacral root stimulation,” IEEE Transactions on Rehabilitation Engineering, 2(2):92 (1994)
Mushahwar V K et al., “Muscle recruitment through electrical stimulation of the lumbo-sacral spinal cord,” IEEE Trans Rehabil Eng, 8(1):22-9 (2000)
Deurloo K E et al., “Transverse tripolar stimulation of peripheral nerve: a modelling study of spatial selectivity,” Med Biol Eng Comput, 36(1):66-74 (1998)
Tarver W B et al., “Clinical experience with a helical bipolar stimulating lead,” Pace, Vol. 15, October, Part II (1992)
In physiological muscle contraction, nerve fibers are recruited in the order of increasing size, from smaller-diameter fibers to progressively larger-diameter fibers. In contrast, artificial electrical stimulation of nerves using standard techniques recruits fibers in a larger- to smaller-diameter order, because larger-diameter fibers have a lower excitation threshold. This unnatural recruitment order causes muscle fatigue and poor force gradation. Techniques have been explored to mimic the natural order of recruitment when performing artificial stimulation of nerves to stimulate muscles.
Fitzpatrick et al., in “A nerve cuff design for the selective activation and blocking of myelinated nerve fibers,” Ann. Conf. of the IEEE Eng. in Medicine and Biology Soc, 13(2), 906 (1991), which is incorporated herein by reference, describe a tripolar electrode used for muscle control. The electrode includes a central cathode flanked on its opposite sides by two anodes. The central cathode generates action potentials in the motor nerve fiber by cathodic stimulation. One of the anodes produces a complete anodal block in one direction so that the action potential produced by the cathode is unidirectional. The other anode produces a selective anodal block to permit passage of the action potential in the opposite direction through selected motor nerve fibers to produce the desired muscle stimulation or suppression.
The following articles, which are incorporated herein by reference, may be of interest:
Rijkhoff N J et al., “Orderly recruitment of motoneurons in an acute rabbit model,” Ann. Conf. of the IEEE Eng., Medicine and Biology Soc., 20(5):2564 (1998)
Rijkhoff N J et al., “Selective stimulation of small diameter nerve fibers in a mixed bundle,” Proceedings of the Annual Project Meeting Sensations/Neuros and Mid-Term Review Meeting on the TMR-Network Neuros, Apr. 21-23, 1999, pp. 20-21 (1999)
Baratta R et al., “Orderly stimulation of skeletal muscle motor units with tripolar nerve cuff electrode,” IEEE Transactions on Biomedical Engineering, 36(8):836-43 (1989)
The following articles, which are incorporated herein by reference, describe techniques using point electrodes to selectively excite peripheral nerve fibers distant from an electrode without exciting nerve fibers close to the electrode:
Grill W M et al., “Inversion of the current-distance relationship by transient depolarization,” IEEE Trans Biomed Eng, 44(1):1-9 (1997)
Goodall E V et al., “Position-selective activation of peripheral nerve fibers with a cuff electrode,” IEEE Trans Biomed Eng, 43(8):851-6 (1996)
Veraart C et al., “Selective control of muscle activation with a multipolar nerve cuff electrode,” IEEE Trans Biomed Eng, 40(7):640-53 (1993)